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HAL Id: inria-00070262

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in fMRI

Thomas Deneux, Olivier Faugeras

To cite this version:

Thomas Deneux, Olivier Faugeras. Parameter estimation efficiency using nonlinear models in fMRI.

[Research Report] RR-5758, INRIA. 2006, pp.35. �inria-00070262�

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ISRN INRIA/RR--5758--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème BIO

Parameter estimation efficiency using nonlinear models in fMRI

Thomas Deneux — Olivier Faugeras

N° 5758

Novembre 2005

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Thomas Deneux , Olivier Faugeras

Thème BIO — Systèmes biologiques Projets Odyssée

Rapport de recherche n°5758 — Novembre 2005 — 35 pages

Abstract: There is an increasing interest in using physiologically plausible models in fMRI analysis. These models do raise new mathematical problems in terms of parameters estima- tion and interpretation of the measured data.

We present some theoretical contributions in this area, using different variations of the Balloon Model (Buxton et al., 1998; Friston et al., 2000; Buxton et al., 2004) as example models. We propose 1) a method to analyze the models dynamics and their stability around equilibrium, 2) a new way to derive least square energy gradient for parameter estimation, 3) a quantitative measurement of parameter estimation efficiency, and 4) a statistical test for detecting voxel activations.

We use these methods in a visual perception checker-board experiment. It appears that the different hemodynamic models considered better capture some features of the response than linear models. In particular, they account for small nonlinearities observed for stim- ulation durations between 1 and 8 seconds. Nonlinearities for stimulation shorter than one second can also be explained by a neural habituation model (Buxton et al., 2004), but fur- ther investigations should assess whether they are rather not due to nonlinear effects of the flow response.

Moreover, the tools we have developed prove that statistical methods that work well for the GLM can be nicely adapted to nonlinear models. The activation maps obtained in both frameworks are comparable.

Key-words: Nonlinear hemodynamic, Balloon Model, system identification

thomas.deneux@ens.fr

olivier.faugeras@sophia.inria.fr

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de paramètres et applications

Résumé : L’utilisation de modèles physiologiquement plausible dans l’analyse des données d’IRM fonctionnelle connaît un intérêt grandissant. Ces modèles soulèvent de nouveaux pro- blèmes mathématiques en ce qui concerne l’estimation de leur paramètres et l’interprétation des données.

Nous présentons des contributions théroriques dans ce domaine, en utilisant plusieurs variations du "Balloon Model" (Buxton et al., 1998; Friston et al., 2000; Buxton et al., 2004). Nous développons 1) une méthode pour analyser les dynamisques des modèles, et leur stabilité autour de l’équilibre, 2) une nouvelle manière de calculer le gradient de l’énergie des moindres carrés utilisée dans l’estimation des paramètres, 3) une mesure quantitative de la précision de cette estimation des paramètres, et 4) un test statistique pour détecter l’activation voxel par voxel.

Nous utilisons ces méthodes pour l’analyse d’une expérience de perception. Les modèles hémodynamiques considérés sont apparus capables de rendre compte de certains aspects de la réponse que les modèles linéaires ignoraient. En particulier, les non-linéarités observées pour des stimulations de une à huit secondes. Les non-linéarités observées pour des stimula- tions plus courtes ont également pu être expliquées par un modèle d’habituation neuronale (Buxton et al., 2004), mais nous nous demandons si elles n’ont pas lieu en réalité dans la réponse du flux sanguin.

Les outils que nous avons développés prouvent que les méthodes statistiques couramment utilisée dans le cadre du Modèle Linéaire Général (GLM) peuvent ête adaptées aux modèles non-linéaires. Les cartes d’activation obtenues avec les deux approches sont en réalité très similaires.

Mots-clés : modèle hémodynamique, identification de système dynamique non linéaire

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1 Introduction

Most fMRI analyses rely on the hypothesis that the BOLD response is an affine function of the neural activity. This hypothesis allows the use of such powerful tools as linear regressions and statistical tests (Friston et al., 1995).

Many studies have considered the question of the range of validity of this linear as- sumption. They all agree on the fact that it holds for stimulation duration or interstimulus intervals (ISI) larger than a threshold. The value of this threshold varies among studies : 2-3 seconds (Boynton et al., 1996; Dale and M., 1997) to 4-6 seconds (Birn et al., 2001; Glover, 1999; Miller et al., 2001; Vazquez and Noll, 1998). Studies involving other measurement modalities established that some nonlinearities in the BOLD were not present at the neural level, and hence were due to hemodynamic effects: blood flow measurement in humans via Arterial Spin Labelling (Miller et al., 2001; Obata et al., 2004), or electrical activity in ani- mals (Janz et al., 2001). Other objections to linearity were raised also, like the apparition of a drift in the BOLD during long stimulations (Krüger et al., 1999).

Besides these experimental observations, there has been a sustained effort to model the long chain that extends between neural activity and the BOLD response: which part of neural activity best correlates with fMRI? (Logothetis and Pfeuffer, 2004), energy consump- tion and metabolic demand (Aubert and Costalat, 2002), blood flow increase signal (Friston et al., 2000; Glover, 1999), vascular mechanic and oxygen extraction (Buxton et al., 1998;

Hoge et al., 2005; Zheng et al., 2002), paramagnetic effect of the deoxyhemoglobin (Ogawa et al., 1993; Davis et al., 1998). A very detailed model can be found in (Aubert and Costalat, 2002), while (Buxton et al., 2004) presents a simplified synthesis.

There have been several attempts to handle nonlinearities in fMRI studies. Some consist in characterizing them as empirical functions of the stimulation patterns, via Volterra kernels (Friston et al., 1998) or specific basis functions (Wager et al., 2005) that could be integrated to the GLM. Others rather bring physiological models in the analysis: they replace linear regression by fitting model output to measured data via parameter estimation.

Thus, Friston and colleagues (Friston et al., 2000) worked with Buxton’s Balloon Model (Buxton et al., 1998), to which they added a damped oscillator in order to model blood flow.

They estimated the model parameters in activated voxels using a Volterra kernel expansion to characterize the model dynamics. Neural signal time courses were approximated by the stimulus up to a scaling factor called neural efficiency, which was estimated as well. Later (2002) they introduced a Bayesian estimation framework that allowed the use of priors on parameters values, didn’t need the Volterra kernels any more, and eventually produced a posteriori probability distributions of the parameters.

Riera et al. (2004) used the same physiological model in a more general framework, allowing noise in the evolution equations in addition to measurement noise. They used a local linearization filter in the spirit of Kalman filter methodology. This filter allowed them to compute an estimation of the system input. This estimation had to lie in a specific vector space, otherwise the problem was underdetermined. This method can be compared to deconvolution tentatives in the linear framework (Glover, 1999).

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We develop our estimation algorithm in the same framework as Friston et al. (2000), i.e.

with a stimulus-locked input to the system, no priors on the parameters, and a Gaussian measurement noise model. These hypotheses allow an easier interpretation and quantifica- tion of the estimation results, since there are only a few indeterminate variables. Estimating the model parameters can be formulated as a simple least square minimization problem. In order to achieve the energy minimization, we propose to compute the system output gradi- ent with respect to the parameters through the integration of a new differential system. It requires no particular form for the input, makes no linear approximation, and the estimation is robust to low frequency drifts in the data.

Once the parameters have been estimated, it is highly important to evaluate the estima- tion accuracy. In effect, it turns out that some parameters are poorly identifiable, because they do not influence much the model output, or because their effect on the ouput interferes with that of other parameters. We propose a sensitivity analysis method, relying on the system output derivative with respect to the parameters, to quantify the identifiability of each parameter. A parallel can be made with the Bayesian framework in (Friston, 2002).

Last, with the interpretation of estimation results the question arises of detecting acti- vations. As proposed by Friston et al. , the efficiency parameter estimated in each voxel is a good candidate to measure activation. But due to identification problems its values do interfere with those of other parameters. So we prefer an activation detection based on statistical significance of the fit between predicted output and measured data. We thus propose a F-test to answer this question.

Before we dive into the details of our contribution, we start with a short analysis of the Balloon Model dynamics.

2 Methods

Physiological models for the BOLD response can be formulated as input-state-output sys- tems (Friston et al., 2000), the input ubeing the stimulus function, the statexbeing a set of non-measurable variables, and the output y being the BOLD signal at the same voxel.

This system is driven by the following evolution and measurement equations:

x(t, u, θ) =˙ F(x(t, u, θ), u(t), θ)

y(t, u, θ) = G(x(t, u, θ), θ), (1)

whereF andGare nonlinear functions, andθrepresents the set of model parameters.

The Balloon Model proposed by Buxton and al. (1997; 1998) and completed by Friston (2002) (flow dynamic) describes the dynamics of the blood flowf, the blood venous volume

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v, the veins deoxyhemoglobin content q (these values are normalized and thus equal 1at rest), and the BOLD signaly:

f¨ = uκsf˙κf(f1)

˙

v = 1τ(fv1/α)

˙

q = 1τ(f1−(1−EE00)1/f v1/α−1q)

y = V0(k1q+k2q/v+k3v)V0(a1(1q)a2(1v)).

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Note that to match the general formulation in (1), it is necessary to add f˙ as a state variable (Friston et al. considered it as a physiological "flow inducing signal"). The system evolution parameters are the neural efficiency, the flow decay κs, the flow time constant κf, the venous transit timeτ, Grub’s parameterα, the oxygen extraction at restE0and the blood volume fraction at restV0; they may vary across brain regions and across subjects.

The measurement parameters a1 and a2 are scanner-dependent. Their values have been evaluated toa1 = 7E0+ 2and a2=2E0+ 2.2for a 1.5 T scanner (Buxton et al., 1998).

But our experimental data was acquired on a 3T scanner, and less is known at this field strength, except that the volume effect a2 is smaller. We useda2 =a1/9and considered the product b = V0(a1+a2) as an unknown quantity, leading to measurement equation y =b(0.9(1q)0.1(1v)). There is too much indetermination in the parameters estimation indeed, to allow us to estimate botha1anda2.

There have been several enhancements of the Balloon Model since then, and Buxton et al. (2004) put them together nicely recently. Three more variables are considered: the metabolism (CMRO2)mbecomes an independent variable instead of the flow-locked expres- sionf1−(1−EE 0)1/f

0 ; the neural activityN is the output of a simple habituation model (with a neural inhibition I) instead of the stimulus-locked expression u. Flow and metabolism are not described by an evolution equations any more, but as convolutions of neural activity with gamma-variate functions:

N = uI I˙ = τ1InNI)

f = 1 + (f11)hf(tδt)N m = 1 + (m11)hm(t)N

˙

v = τ1(f (v1/α+τviscv))˙

˙

q = τ1(mqv(v1/α+τviscv))˙ y = V0(a1(1q)a2(1v)),

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with

( hf(t) = 1f(τtf)3e

t τf

hm(t) = 1m(τtm)3eτmt .

Additional parameters are the inhibitory time constant τI, the inhibitory gain factor κn, the normalized CBF and CMRO2 responses to sustained activityf1 andm1, the delay

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δt between CMRO2 and CBF responses, the widths τf and τm of the CBF and CMRO2 impulse responses, and the volume viscoelastic time constantsτvisc+ andτvisc (an hysteresis rule is authorized for the volume dynamics: the viscosity parameterτvisc can take 2 different values whether ∂v

∂t >0visc =τvisc+ ) or ∂v

∂t <0visc =τvisc )).

We note that the neural habituation model results in N= u hIu, where

hI(t) = κn

τIeκnτI+1t. Thus we can write

f = 1 +ξ(hf(tδt)hf(tδt)hI)u = 1 +ξ Hfu m= 1 +nξ (hmhmhI)u = 1 +nξ Hmu,

whereξ=(f11), andn= (f11)/(m11)is the steady-state flow-metabolism ratio.

In the following, we will estimateξ andninstead of,f1andm1. We also note that volume evolution equation can be transformed in

˙

v= 1

τ+τvisc

(fvα1) = ( 1

τ+τvisc+ (f vα1) iffα> v

1

τ+τvisc (f vα1) iffα< v.

Hence, the new Balloon Model formulation becomes

˙

v = τ+τ1

visc(1 +ξ Hf uv1/α)

˙

q = τ1(1 +nξ Hmuvq(v1/α+τviscv))˙ y = V0(a1(1q)a2(1v)).

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The tools we develop in the following section will be used with the two models ((2) and (4)).

2.1 System dynamic and stability

Before analyzing in detail these two models it is interesting to build some intuition for their dynamics. We refer to previous studies for several simulated time courses of state variables and BOLD output (Buxton et al., 1998; Friston et al., 2000; Riera et al., 2004). Figure 1 demonstrates nonlinear effects of the initial Balloon Model in the response peak amplitudes by comparing responses to given stimulation lengths to their prediction from responses to shorter stimulations.

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0 10 20 30

−5 0 5 10x 10−3

0.25s/0.5s

0 10 20 30

−5 0 5 10x 10−3

0.5s/1s

0 10 20 30

−0.01 0 0.01 0.02

1s/2s

0 10 20 30

−0.02 0 0.02 0.04

2s/4s

0 10 20 30

−0.02 0 0.02 0.04

4s/8s

0 10 20 30

−0.02 0 0.02 0.04

8s/16s

Figure 1: Balloon model simulation for increasing stimulation lengths and visualization of nonlinearities. The response to stimulation length2T is compared to the sum of 2 shifted re- sponses to stimulation lengthT. Nonlinearities are stronger for 2s/4s and 4s/8s comparisons (= 0.4,κs= 0.65,κf = 0.4,τ = 1,α= 0.4,E0= 0.4,V0(a1+a2) = 0.1).

The hemodynamic main effect seems to be roughly a smoothing of the input, and it looks unlikely that any special dynamics like bifurcations, limit cycles... can occur. Indeed we prove that for a constant inputu0, there is only one stable equilibrium point.

Let us consider here the first Balloon Model formulation (2). The flow dynamic equation is a pure linear damped oscillator. It can then be computed exactly by a convolution

f(t) = 1 +ku(t).

If we assume∆ =κ2sf>0, we have:

k(t) =eκs2 tcos(

pfκ2s

2 t)

(if we had<0,kwould have been of a different form, with exponentials only).

Since it is a linear convolution, the flow cannot have any special dynamic (if the input is constant the flow converges necessarily to the equilibrium point1 +u0f).

One remark is that sincek(t)<0for some t, f can theoretically have negative values, even ifu(t)>0, t. Figure 2 shows that we can obtain flow time courses with non-realistic

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0 5 10 15 20 0

2 4 6 8

0 5 10 15 20

0 2 4 6 8

Figure 2: (A) flow time courses for a 4s input (= 0.4,κs= 0.65,κf = 0.4,τ = 1,α= 0.4, E0= 0.4,V0(a1+a2) = 0.1). (B) flow time course becomes unrealistic for= 3!

values, e.g., negative, if the productuis too large (it does not happen in practice since it would require non-realistic values for).

Volume only depends on flow. If we notev(f) =fα,v˙in (2) has the same sign asv(f)v.

The equation looks like an exponential decay to steady state, though it is nonlinear. If the input is constant, the flow and the volume necessarily converge to their equilibrium points (1 +u0f)and(1 +u0f)α.

In a similar way, if we noteq(v, f) =f1−(1−EE00)1/f v1−1/α,q˙is the same sign asq(v, f)q.

If the input is constant the deoxyhemoglobin content eventually converges to an equilibrium point.

We just gave an intuitive proof of the system stability for a constant input. From a more mathematical viewpoint, it is also possible to show it by examining the eigenvalues of the Jacobian of the evolution functionF at the equilibrium pointx0.

x0 is determined by equalling the time derivativeF(x0, u0, θ)to zero:

x0=

0 1 + uκf0 (1 +uκf0)α

1−(1−E0)1/(1+

u0 κf )

E0 (1 + uκf0)α

.

The jacobian ofF atx is:

∂F∂x =

κs κf 0 0

1 0 0 0

0 τ1 x1/α−13

ατ 0

0 ∂F4

∂x2

τ1(1α1)x1/α−23 x4 x1/α−13 τ

,

with

∂F4

∂x2 =τ1(1(1E0)1/x2

E0 log(1E0)(1E0)1/x2 E0x2 ),

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and its eigenvalues evaluated atx0 are {−κs+

κ2 s−4κf

2 ,κs

κ2 s−4κf

2 ,(1+

u0 κf)1−α

ατ ,(1+

u0 κf )1−α

τ }

(they can be obtained as follows: note that the matrix ∂F

∂x is trigonal by blocks with block sizes equal to 2, 1, and 1; the four eigenvalues are respectively the 2 eigenvalues of the first 2x2 block, and the third and fourth diagonal terms).

Since the physiological parameters are always positive, these eigenvalues are either real and negative or have negative real parts: the system is always stable around equilibrium.

Similar considerations (intuitive interpretation of the equations as well as differentiation of the evolution function) do lead to the conclusion of the uniqueness and stability of the equilibrium point in the second Balloon Model formulation.

2.2 Parameter estimation

We model the measured data as the sum:

y=f(u, θ) +e, e∼ N(0,Σ), (5)

wheref(u, θ)is the output of the dynamical system with inputu(stimulus-locked activ- ity) and parametersθ, andeis a Gaussian noise with varianceΣ. This does not necessarily ignore physiological noise: if the nonlinear effects of the model are small enough, and if we note en the cortical noise (ongoing activity) andem the measurement noise, we can make the following approximation:

y = f(u+en, θ) +em

f(u, θ) +f(en, θ) +em=f(u, θ) +e. (6) If en andem are supposed Gaussian, then resulting e is also a Gaussian colored noise.

In fact, we assume in our study a white noise Σ = σ2I. The methods we present can be extended to a colored noiseΣ =σ2Σ0, but it would require to estimate autocorrelations to defineΣ0; this will be discussed later.

Parameter estimation is obtained by maximizing the likelihood of the measured data y with respect to the parametersθ:

θˆ = argmaxθp(y|θ)

= argminθ logp(y|θ)

= argminθ (f(u,θ)−y)TΣ21(f(u,θ)−y). Under the white noise assumption, it leads us to minimize the energy

E(θ) = (f(u, θ)y)T(f(u, θ)y).

To minimize E(θ), we use a Levenberg Marquardt algorithm (Press et al., 1992; Mar- quardt, 1963), implemented in the Matlab function ’lsqcurvefit’. The algorithm needs at

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each iteration step the Jacobian of the system output with respect to the parameters ∂f

∂θ. As predicted state and outputx(t, u, θ)andy(t, u, θ)are defined by a differential system, it is possible to define ∂x

∂θ(t, u, θ)and ∂y

∂θ(t, u, θ)with a new differential system .

Let us go back to the initial system (1) indeed (for clarity, we use here x instead of x(t, u, θ)):

x˙ = F(x, u(t), θ) y = G(x, θ).

Differentiating both sides of these equations with respect toθ, we get the new system

∂x˙

∂θ = ∂F

∂x(x, u(t), θ)∂x

∂θ +∂F

∂θ(x, u(t), θ)

∂y

∂θ = ∂G

∂x(x, θ)∂x

∂θ +∂G

∂θ(x, θ).

This system can be integrated numerically, using the initial conditions ∂x

∂θ(t = 0) = 0 (at time t = 0, the state variables are at rest and do not depend upon θ). We therefore obtain the numerical values of ∂f

∂θ = (∂y

∂θ(t, u, θ))0≤t≤T without using any linearization of the system of equations.

2.3 Handling confounds effects

It is often usefull to ignore a certain set of timecourse components in real datasets, low frequencies for example.

Let us note C the matrix whose columns are the undesirable components. ThenpC = IC(CTC)−1CT is the projector orthogonal to these confounds. Ignoring them consists in fittingpCf(u, θ)topCy instead of fittingf(u, θ)to y.

The new energy writes

E(θ) = (pC(f(u, θ)y))T(pC(f(u, θ)y)) = (f(u, θ)y)TpC(f(u, θ)y), and the gradient ofpCf(u, θ) against parameters for using Levenberg-Marquardt algo- rithm ispC∂f

∂θ(u, θ), with ∂f

∂θ(u, θ)computed as previously.

2.4 Sensitivity analysis

A serious obstacle to parameter estimation is the identifiability of the system, i.e. do we have enough information once we know the system input uand output y to determine the parameter values ? Is there a unique solutionθ to the equationy=f(u, θ)?

Actually this is hardly the case for the Balloon Model, because the effects of some parameters on the output do interfere with those of others. The extreme case would be for

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example if the scale factor on the input (neural efficacy) and that on the output (V0) were estimated on data with an input low enough to make the linear approximation of the model hold. Indeed, increasing the first could be compensated by decreasing the second by the same factor to produce exactly the same output. It would not be possible to estimate these 2 parameters, but only their product.

We want to investigate how much the system output is sensitive to changes in one parameter. Let us noteθithis parameter, andθ2the rest of parameters (θ={θi, θ2}). Also we noteJ = ∂f

∂θ(u, θ) the Jacobian of system output,Ji itsithcolumn and J2 the matrix consisting of the remaining columns. For a small parameter changewe have

f(u, θ+dθ) =f(u, θ) +Jdθ=f(u, θ) +Jii+J22.

For a small changeiofθi,fvaries byJii; however, ifJiis not orthogonal to the other Jacobian componentsJ2, part of this variation can be compensated for by a change in the other parameters: 2=J2+Jii, whereJ2+= (J2TJ2)−1J2T denotes the pseudo-inverse of J2. We then have:

min2 kf(u, θ+dθ)f(u, θ)k=k(IJ2J2+)Jiik=πi|i|, with

πi=k(IJ2J2+)Jik= q

JiT(IJ2J2+)Ji.

The bigger πi, the more identifiable θi is. This also means that, for a given percent- age x, if θi changes by less than πi−1xkf(u, θ)k, one can adjust the other parameters θ2 to make the model output vary by less than x%. Given an input u and an initial parameter set θ0, our sensitivity analysis consists in considering the sensitivity intervals 0iπ−1i xkf(u, θ0)k, θ0i+π−1i xkf(u, θ0)k], with x = 1 to 5%. They are not confidence intervals for parameter estimation! Rather they indicate that the system output is very little sensitive to changes ofθi inside these intervals.

Figure 3 shows such a sensitivity analysis withx= 1%for two different inputs (a single impulse and two successive impulses). The sensitivity intervals are represented in the left column of the figure. For each of the seven parameters (encoded with different colors) we fix it to one of the two bounds of its sensitivity interval compute the values of the other six parameters from2=J2+Jii and plot the resulting time course. The figure clearly shows that very different parameter sets can result in very similar system outputs (table 2.4 shows the obtained parameter sets and the output variations). It also appears that the sensitivity depends on the input complexity: in the second case the parameters are more identifiable, because the effects of the different parameters can be more diverse and hence less correlated. For that reason, the experimental design we present later uses a large panel of ISI and stimulus duration to increase identifiability.

As a final word of caution, be aware that we have only discussed identifiability at a local scale, i.e. we only considered one minimum of the energy and approximated the shape of

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0 0.5 1 1.5 2 2.5 3 3.5

0 0.5 1 1.5 2 2.5 3 3.5

0 2 4 6 8 10 12 14 16 18 20

−2 0 2 4 6 8 10x 10−3

0 2 4 6 8 10 12 14 16 18 20

−2 0 2 4 6 8 10 12x 10−3

2.5 3 3.5 4 4.5

8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 10 x 10−3

2.5 3 3.5 4 4.5 5 5.5

8 8.2 8.4 8.6 8.8 9 9.2 9.4 9.6 9.8 x 10−3

κs

κf

τ α

E0

V0(a1+a2) α τ κf

κs

E0

V0(a1+a2)

Figure 3: Sensitivity analysis for two different system inputs around a given θ0. Top:

response to an impulse. Bottom: response to 2 consecutive impulses. Left: 1% signal change sensitivity intervals - color stars show different parameter sets with one parameter constrained to be at the edge of its sensitivity interval (e.g., red corresponds tobeing fixed and the other parameters computed with 2 = J2+Jii). Right: output variations for these parameter sets compared to the output for the referenceθ0 (bold dashed line). Values of all parameters and percentages of signal changes are given in table 2.4.

possible model outputs locally with the tangent plane (see figure (4)). But since the shape can be more complicated, indetermination can be even worse than the one resulting from the above discussion.

2.5 A Bayesian formulation of sensitivity

The above results can be related to Bayesian inference principles. Let us noteθ0 the true parameter set and approximatef atθ0 with its first order Taylor series expansion

f(u, θ) =f(u, θ0) +Jθ0θ0). (7) We recall the probability model for measured data (5):

y=f(u, θ) +e, e∼ N(0, σ2I).

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parameter κs κf τ α E0 b % output change

θ0 1 .65 .4 1 .4 .4 .1

fixed 1.56 .66 .43 1.2 .33 .46 .07 1%

κs fixed 1 .69 .41 1.01 .39 .4 .1 2.1%

κf fixed 1.2 .64 .42 1.26 .23 .55 .08 1%

τ fixed 1.15 .7 .47 1.32 .34 .36 .09 3.5%

αfixed 1.25 .68 .42 0.86 .7 .28 .15 2%

E0 fixed 1.52 .56 .37 0.89 .27 .76 .09 5.6%

a1fixed 0.43 .64 .4 1.07 .31 .35 .18 1.3%

parameter κs κf τ α E0 b % output change

θ0 1 .65 .4 1 .4 .4 .1

fixed 1.33 .67 .43 1.13 .37 .43 .08 1.29%

κs fixed 1 .67 .4 0.99 .4 .4 .1 1%

κf fixed 1 .67 .42 1.03 .4 .4 .1 0.8%

τ fixed 1.12 .69 .47 1.22 .39 .35 .1 2.4%

αfixed 0.97 .66 .41 0.95 .57 .22 .12 1.1%

E0 fixed 1.24 .64 .39 0.93 .3 .72 .12 3.9%

a1fixed 0.67 .65 .39 0.97 .35 .48 .15 0.9%

Table 1: Parameter values for the sensitivity analysis in figure 3: quite different parameter sets can lead to very similar system outputs. The output variation is not exactly1%when one parameter is fixed to the edge of its sensitivity interval, because these intervals were calculated using first order approximations with respect to parameters.

We do not use an a priori Gaussian distribution for the parameters as Friston (2002), but a non-informative degenerate uniform distributionθ∼ U(R). See (Kershaw et al., 1999) for using such methods in fMRI data analysis and (Box and Tiao, 1992) for more theoretical details.

Then we can calculate the a posteriori distribution for parameterθusing Bayesian inference p(θ|y)p(y|θ)p(θ),

which results in a Gaussian distribution with mean and variance (see Appendix for detail) E(θ) = θ0+ (JTJ)−1JT(yf0))

V(θ) = (JTJ)−1. (8)

The variance of parameterθi is theithdiagonal term inV(θ): (JTJ)−1ii .

It can also be calculated by forming the marginal a posteriori distribution of parameter θi

p(θi|y) = Z

p(θi, θ2|y)dθ2.

(17)

We show by integrating this formula (in appendix) that the a posteriori variance is equal to σ2π−2i . This shows that the incertitude in θi estimation we established above is proportional to its a posteriori variance when there is a Gaussian white measure noise, and it also gives us the new formula

π−1i = q

(JtJ)−1ii .

However we have observed in simulation and data (not shown) that actual variance ofθi

is more thanσ2π−2i . This is probably due to the linearization and white noise assumption.

This is the reason why we prefer incertitude intervals as defined previously to statistical confidence intervals.

2.6 Statistical test

We want to establish a statistical test to detect activations voxelwise. We use the presented bayesian framework (but since we do not know the true parameter setθ0, we use our estima- tionθˆinstead). In particular, we still use the linear approximation above (7), since it is too difficult to establish probabilities on parameters in the nonlinear case. This actually means that we approximate the manifold of all possible model outputs by its tangent vectoriel subspace at pointfθ)(figure 4A).

Friston (2002) proposed in a similar bayesian framework an estimation detection based on the marginal distribution of neural efficiency parameter. However, it can happen that this marginal distribution is pretty flat and says ’= 0is plausible’, not because there is no detected activation in the data, but only becauseis poorly identifiable, due to interactions with other parameters (see figure 4D-E). For that reason, we would prefer a test based on the whole set of parameters, or equivalently on the model fit to data, by calculating how probable it is thatf(θ) = 0(i.e. θ= ˆθJ+fθ)).

We cannot establish a statistical test directly from θ Gaussian a posteriori distribu- tion calculated above (8), since variance parameterσ2 is unknown. Again, we use a non- informative degenerate a priori distribution for σ2, p(σ2) σ12. Then we can derive (see Appendix) a new a posteriori distribution forθ, that follows a Student law withν= (np) degrees of freedom (n and pbeing the numbers of measure point and parameters, respec- tively), and with mean and variance

E(θ) = θˆ

V(θ) = σˆ2(JTJ)−1, (9)

where

ˆ

σ2 = n−p1 (yfθ))T(yfθ)).

If we used a colored version of noiseΣ =σ2Σ0, a similar distribution could be obtained, but the number of degrees of freedom ν would depend on the rank of Σ0 (Friston and Worsley, 1995).

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